On Sets Defining Few Ordinary Lines

  title={On Sets Defining Few Ordinary Lines},
  author={Ben Green and Terence Tao},
  journal={Discrete \& Computational Geometry},
  • B. Green, T. Tao
  • Published 23 August 2012
  • Mathematics
  • Discrete & Computational Geometry
Let $$P$$P be a set of $$n$$n points in the plane, not all on a line. We show that if $$n$$n is large then there are at least $$n/2$$n/2ordinary lines, that is to say lines passing through exactly two points of $$P$$P. This confirms, for large $$n$$n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $$n-C$$n-C ordinary lines for some absolute constant $$C$$C. We also solve, for large $$n$$n, the “orchard… 
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